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様子 : Most informal, most 'soft'. Most heavily used in colloquial situations. Suitable for insignificant things (as in 焼き芋の様子はどうだ？), but also for significant things (as in 洋子の手術後の様子はどうだ？).

状況 : A bit formal, but also frequently used in colloquial situation. Mostly used for things that can be observed externally (as opposed to a person's feelings etc.). For example, 事故の状況. If you say 洋子の手術後の状況, the listener will think of heart beat, blood pressure, breath rate, the doctor's assessment etc. If you say 洋子の手術後の様子, the listener will think of how she feels now, whether she is feeling pain etc.

状態 : Similarly to 状況, it's used to signal objectivity. However, it can also be used for things that cannot be observed from outside like feelings. For example, if you'd say 洋子の手術後の心の状態, the speaker is indicating that he/she is making an objective observation rather than just expressing empathy. If you'd say 洋子の手術後の心の様子, there is much more empathy. Also, more formal than 様子 but is also used heavily in colloquial situation. Also mostly used for static things. 状況 or 様子 can be used for things that are changing right now, but 状態 is more suitable for static things. For example, if you'd say 株価の状況, the listener will think of the prices changing minute by minute. If you say 株価の状態, he/she will think of a more long timespan like this year, or this month. At minimum the image they get will be a more static price rather than actively changing prices.

事情 : Mostly to talk about reasons, as in 風邪で動けなかったという事情を酌んでいただき、許しては頂けないでしょうか？ However, can also be used similarly to 状況, as in 経済事情. There is a difference between 状況 and 事情 though. If there is a report about 経済状況, the listener would think of a summary of the current situation or something. If you talk about 経済事情, it kinda sound more deep and the listener would anticipate a more deep report/analysis than 経済状況.